a. find the slope of the tangent line to the graph of f at the given point. b. find the slope - intercept…

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=2\sqrt{x}$ as $f(x) = 2x^{\frac{1}{2}}$.

Step2: Find the derivative

Use the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$. So $f^\prime(x)=2\times\frac{1}{2}x^{\frac{1}{2}-1}=x^{-\frac{1}{2}}=\frac{1}{\sqrt{x}}$.

Step3: Evaluate the derivative at the given point

Substitute $x = 1$ into $f^\prime(x)$. $f^\prime(1)=\frac{1}{\sqrt{1}}=1$. This is the slope of the tangent line.

Step4: Find the equation of the tangent line

Use the point - slope form $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(1,2)$ and $m = 1$. So $y - 2=1\times(x - 1)$.

Step5: Rewrite in slope - intercept form

Expand and simplify: $y-2=x - 1$, then $y=x+1$.

Answer:

a. 1 b. $y=x + 1$